How to rotate object axis, leaving the object intact?
I propose another approach, trying to simplify your issue.

It uses curves and hook:



How to set it up:


Create a segment (starting from a plane)
Place each extremity to the center of spheres
Turn it to curve AltC
Hook CtrlH each to the corresponding sphere


The hook works like this:


Select the object to hook to
Shift select the object to hook
Enter edit mode
Select the vertex to hook
Hook CtrlH
And choose "to selected object"




After that, depending on your needs, you can turn the curves back to meshes or keep them and animate using "start" or "end" bevel options in the curve properties panel (that can replace the shape key animation you talked about).